(* Inversion lemmas with restricted rt-transition for terms *****************)
lemma cnv_cpr_tdeq_fwd_refl (a) (h) (G) (L):
- ∀T1,T2. ⦃G, L⦄ ⊢ T1 ➡[h] T2 → T1 ≛ T2 →
- ⦃G, L⦄ ⊢ T1 ![a,h] → T1 = T2.
+ ∀T1,T2. ⦃G,L⦄ ⊢ T1 ➡[h] T2 → T1 ≛ T2 →
+ ⦃G,L⦄ ⊢ T1 ![a,h] → T1 = T2.
#a #h #G #L #T1 #T2 #H @(cpr_ind … H) -G -L -T1 -T2
[ //
| #G #K #V1 #V2 #X2 #_ #_ #_ #H1 #_ -a -G -K -V1 -V2
qed-.
lemma cpm_tdeq_inv_bind_sn (a) (h) (n) (p) (I) (G) (L):
- ∀V,T1. ⦃G, L⦄ ⊢ ⓑ{p,I}V.T1 ![a,h] →
- ∀X. ⦃G, L⦄ ⊢ ⓑ{p,I}V.T1 ➡[n,h] X → ⓑ{p,I}V.T1 ≛ X →
- ∃∃T2. ⦃G,L⦄ ⊢ V ![a,h] & ⦃G, L.ⓑ{I}V⦄ ⊢ T1 ![a,h] & ⦃G, L.ⓑ{I}V⦄ ⊢ T1 ➡[n,h] T2 & T1 ≛ T2 & X = ⓑ{p,I}V.T2.
+ ∀V,T1. ⦃G,L⦄ ⊢ ⓑ{p,I}V.T1 ![a,h] →
+ ∀X. ⦃G,L⦄ ⊢ ⓑ{p,I}V.T1 ➡[n,h] X → ⓑ{p,I}V.T1 ≛ X →
+ ∃∃T2. ⦃G,L⦄ ⊢ V ![a,h] & ⦃G,L.ⓑ{I}V⦄ ⊢ T1 ![a,h] & ⦃G,L.ⓑ{I}V⦄ ⊢ T1 ➡[n,h] T2 & T1 ≛ T2 & X = ⓑ{p,I}V.T2.
#a #h #n #p #I #G #L #V #T1 #H0 #X #H1 #H2
elim (cpm_inv_bind1 … H1) -H1 *
[ #XV #T2 #HXV #HT12 #H destruct
lemma cpm_tdeq_inv_appl_sn (a) (h) (n) (G) (L):
∀V,T1. ⦃G,L⦄ ⊢ ⓐV.T1 ![a,h] →
∀X. ⦃G,L⦄ ⊢ ⓐV.T1 ➡[n,h] X → ⓐV.T1 ≛ X →
- ∃∃m,q,W,U1,T2. yinj m < a & ⦃G,L⦄ ⊢ V ![a,h] & ⦃G, L⦄ ⊢ V ➡*[1,h] W & ⦃G, L⦄ ⊢ T1 ➡*[m,h] ⓛ{q}W.U1
- & ⦃G,L⦄⊢ T1 ![a,h] & ⦃G, L⦄ ⊢ T1 ➡[n,h] T2 & T1 ≛ T2 & X = ⓐV.T2.
+ ∃∃m,q,W,U1,T2. yinj m < a & ⦃G,L⦄ ⊢ V ![a,h] & ⦃G,L⦄ ⊢ V ➡*[1,h] W & ⦃G,L⦄ ⊢ T1 ➡*[m,h] ⓛ{q}W.U1
+ & ⦃G,L⦄⊢ T1 ![a,h] & ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 & T1 ≛ T2 & X = ⓐV.T2.
#a #h #n #G #L #V #T1 #H0 #X #H1 #H2
elim (cpm_inv_appl1 … H1) -H1 *
[ #XV #T2 #HXV #HT12 #H destruct
qed-.
lemma cpm_tdeq_inv_cast_sn (a) (h) (n) (G) (L):
- ∀U1,T1. ⦃G, L⦄ ⊢ ⓝU1.T1 ![a,h] →
- ∀X. ⦃G, L⦄ ⊢ ⓝU1.T1 ➡[n,h] X → ⓝU1.T1 ≛ X →
+ ∀U1,T1. ⦃G,L⦄ ⊢ ⓝU1.T1 ![a,h] →
+ ∀X. ⦃G,L⦄ ⊢ ⓝU1.T1 ➡[n,h] X → ⓝU1.T1 ≛ X →
∃∃U0,U2,T2. ⦃G,L⦄ ⊢ U1 ➡*[h] U0 & ⦃G,L⦄ ⊢ T1 ➡*[1,h] U0
- & ⦃G, L⦄ ⊢ U1 ![a,h] & ⦃G, L⦄ ⊢ U1 ➡[n,h] U2 & U1 ≛ U2
- & ⦃G, L⦄ ⊢ T1 ![a,h] & ⦃G, L⦄ ⊢ T1 ➡[n,h] T2 & T1 ≛ T2 & X = ⓝU2.T2.
+ & ⦃G,L⦄ ⊢ U1 ![a,h] & ⦃G,L⦄ ⊢ U1 ➡[n,h] U2 & U1 ≛ U2
+ & ⦃G,L⦄ ⊢ T1 ![a,h] & ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 & T1 ≛ T2 & X = ⓝU2.T2.
#a #h #n #G #L #U1 #T1 #H0 #X #H1 #H2
elim (cpm_inv_cast1 … H1) -H1 [ * || * ]
[ #U2 #T2 #HU12 #HT12 #H destruct
qed-.
lemma cpm_tdeq_inv_bind_dx (a) (h) (n) (p) (I) (G) (L):
- ∀X. ⦃G, L⦄ ⊢ X ![a,h] →
- ∀V,T2. ⦃G, L⦄ ⊢ X ➡[n,h] ⓑ{p,I}V.T2 → X ≛ ⓑ{p,I}V.T2 →
- ∃∃T1. ⦃G,L⦄ ⊢ V ![a,h] & ⦃G, L.ⓑ{I}V⦄ ⊢ T1 ![a,h] & ⦃G, L.ⓑ{I}V⦄ ⊢ T1 ➡[n,h] T2 & T1 ≛ T2 & X = ⓑ{p,I}V.T1.
+ ∀X. ⦃G,L⦄ ⊢ X ![a,h] →
+ ∀V,T2. ⦃G,L⦄ ⊢ X ➡[n,h] ⓑ{p,I}V.T2 → X ≛ ⓑ{p,I}V.T2 →
+ ∃∃T1. ⦃G,L⦄ ⊢ V ![a,h] & ⦃G,L.ⓑ{I}V⦄ ⊢ T1 ![a,h] & ⦃G,L.ⓑ{I}V⦄ ⊢ T1 ➡[n,h] T2 & T1 ≛ T2 & X = ⓑ{p,I}V.T1.
#a #h #n #p #I #G #L #X #H0 #V #T2 #H1 #H2
elim (tdeq_inv_pair2 … H2) #V0 #T1 #_ #_ #H destruct
elim (cpm_tdeq_inv_bind_sn … H0 … H1 H2) -H0 -H1 -H2 #T0 #HV #HT1 #H1T12 #H2T12 #H destruct
Q (L.ⓑ{I}V) T1 T2 → Q L (ⓑ{p,I}V.T1) (ⓑ{p,I}V.T2)
) →
(∀m. yinj m < a →
- ∀L,V. ⦃G,L⦄ ⊢ V ![a,h] → ∀W. ⦃G, L⦄ ⊢ V ➡*[1,h] W →
- ∀p,T1,U1. ⦃G, L⦄ ⊢ T1 ➡*[m,h] ⓛ{p}W.U1 → ⦃G,L⦄⊢ T1 ![a,h] →
- ∀T2. ⦃G, L⦄ ⊢ T1 ➡[n,h] T2 → T1 ≛ T2 →
+ ∀L,V. ⦃G,L⦄ ⊢ V ![a,h] → ∀W. ⦃G,L⦄ ⊢ V ➡*[1,h] W →
+ ∀p,T1,U1. ⦃G,L⦄ ⊢ T1 ➡*[m,h] ⓛ{p}W.U1 → ⦃G,L⦄⊢ T1 ![a,h] →
+ ∀T2. ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 → T1 ≛ T2 →
Q L T1 T2 → Q L (ⓐV.T1) (ⓐV.T2)
) →
(∀L,U0,U1,T1. ⦃G,L⦄ ⊢ U1 ➡*[h] U0 → ⦃G,L⦄ ⊢ T1 ➡*[1,h] U0 →
- ∀U2. ⦃G, L⦄ ⊢ U1 ![a,h] → ⦃G, L⦄ ⊢ U1 ➡[n,h] U2 → U1 ≛ U2 →
- ∀T2. ⦃G, L⦄ ⊢ T1 ![a,h] → ⦃G, L⦄ ⊢ T1 ➡[n,h] T2 → T1 ≛ T2 →
+ ∀U2. ⦃G,L⦄ ⊢ U1 ![a,h] → ⦃G,L⦄ ⊢ U1 ➡[n,h] U2 → U1 ≛ U2 →
+ ∀T2. ⦃G,L⦄ ⊢ T1 ![a,h] → ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 → T1 ≛ T2 →
Q L U1 U2 → Q L T1 T2 → Q L (ⓝU1.T1) (ⓝU2.T2)
) →
∀L,T1. ⦃G,L⦄ ⊢ T1 ![a,h] →
(* Advanced properties with restricted rt-transition for terms **************)
lemma cpm_tdeq_free (a) (h) (n) (G) (L):
- ∀T1. ⦃G, L⦄ ⊢ T1 ![a,h] →
- ∀T2. ⦃G, L⦄ ⊢ T1 ➡[n,h] T2 → T1 ≛ T2 →
- ∀F,K. ⦃F, K⦄ ⊢ T1 ➡[n,h] T2.
+ ∀T1. ⦃G,L⦄ ⊢ T1 ![a,h] →
+ ∀T2. ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 → T1 ≛ T2 →
+ ∀F,K. ⦃F,K⦄ ⊢ T1 ➡[n,h] T2.
#a #h #n #G #L #T1 #H0 #T2 #H1 #H2
@(cpm_tdeq_ind … H0 … H1 H2) -L -T1 -T2
[ #I #L #H #F #K destruct //
(* Advanced inversion lemmas with restricted rt-transition for terms ********)
lemma cpm_tdeq_inv_bind_sn_void (a) (h) (n) (p) (I) (G) (L):
- ∀V,T1. ⦃G, L⦄ ⊢ ⓑ{p,I}V.T1 ![a,h] →
- ∀X. ⦃G, L⦄ ⊢ ⓑ{p,I}V.T1 ➡[n,h] X → ⓑ{p,I}V.T1 ≛ X →
- ∃∃T2. ⦃G,L⦄ ⊢ V ![a,h] & ⦃G, L.ⓑ{I}V⦄ ⊢ T1 ![a,h] & ⦃G, L.ⓧ⦄ ⊢ T1 ➡[n,h] T2 & T1 ≛ T2 & X = ⓑ{p,I}V.T2.
+ ∀V,T1. ⦃G,L⦄ ⊢ ⓑ{p,I}V.T1 ![a,h] →
+ ∀X. ⦃G,L⦄ ⊢ ⓑ{p,I}V.T1 ➡[n,h] X → ⓑ{p,I}V.T1 ≛ X →
+ ∃∃T2. ⦃G,L⦄ ⊢ V ![a,h] & ⦃G,L.ⓑ{I}V⦄ ⊢ T1 ![a,h] & ⦃G,L.ⓧ⦄ ⊢ T1 ➡[n,h] T2 & T1 ≛ T2 & X = ⓑ{p,I}V.T2.
#a #h #n #p #I #G #L #V #T1 #H0 #X #H1 #H2
elim (cpm_tdeq_inv_bind_sn … H0 … H1 H2) -H0 -H1 -H2 #T2 #HV #HT1 #H1T12 #H2T12 #H
/3 width=5 by ex5_intro, cpm_tdeq_free/